The solutions of the Laplace equation $\Delta f =0$ on a domain $D\subset \mathbb{R}^n$ are known as *harmonic functions*.

learn more… | top users | synonyms (1)

2
votes
1answer
47 views

Harmonic conjugates on annulus slit

Let $D$ be an annulus slit with $$D= \{a<|z|< b \}$$ excluding $(-b,-a)$. Show that any harmonic function on $D$ has a harmonic conjugate on $D$. The hint says to fix $c$ between $a$ and ...
0
votes
1answer
45 views

What is a harmonic conjugate of $u=Arg(z)$?

Let $u=Arg(z)$ be a function maps $\mathbb{C}\setminus \{0\}$ to $ (-\pi,\pi]$. How do i find a harmonic conjugate of $u$ when $Arg(z)\in (-\pi,\pi)$?
0
votes
1answer
37 views

Extending bounded functions on $\mathbb{R}$ to $\mathbb{H}$ with the Poisson kernel.

Let $h(\phi):\mathbb{R}\to\mathbb{R}$ be a bounded piecewise continuous function on the real line. Define a function on the upper half-plane by the formula $\tilde{h}(s+it):=\int_{-\infty}^{\infty} ...
0
votes
0answers
20 views

Looking at Green's Function for the Dirichlet Problem, how do you calculate n(y)

If G(x,y) = (-1/4pi)[(|x-y|^-1)-(|x-r(y)|^-1)] where r(y)=(y1,y2,-y3) and x and y are vectors in R3. The question states 'Evaluate ∂G(x,y)/∂ny (n subscript y) for y ∈ ∂Ω. I know to do this you ...
0
votes
0answers
29 views

Analytic property of harmonic functions through mean value property.

I have this strong intuition that one should be able to at least prove that a harmonic function is infinitely differentiable, let alone analytic, using just the mean value property. Unfortunately I am ...
1
vote
0answers
82 views

Show Green's function solves Poisson's equation

I am reading Walter Strauss's book Introduction to PDE. The definition of Green's function is as follows: The Green's function $G(x)$ for the operator $-\Delta$ and the domain $D \subset ...
1
vote
2answers
50 views

Intuitive significance of harmonicity

I'm nearing the end of the semester of an introductory-level complex variables class. (Very introductory -- it's the version of the class that's only required for engineering and physics majors, as it ...
1
vote
1answer
44 views

Need help about harmonic functions!

I have trouble on solving the following problem: Show that there doesn't exist a non-constant function $u$ such that $u$ is harmonic on C and for $z=x+iy$ in C that $u(z)>4x^2+9y^2+1$.
1
vote
2answers
43 views

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic?

Given f(z) is analytic in Domain D, is Arg|f(z)| harmonic? If yes, in which domain?
0
votes
0answers
31 views

Chain Rule (applied twice) for vector valued functions

I need to show that if $f: \mathbb{R}^2 \to \mathbb{R}$ is harmonic, i.e., $$\frac{\partial^2 f}{\partial x^2}(x,y) + \frac{\partial^2 f}{\partial y^2}(x,y) = 0 \quad \text{ for all } (x,y) \in ...
0
votes
1answer
32 views

Existence of solutions to Laplace's equation for almost everywhere smooth boundary conditions

Let $\Omega$ be a compact region in the plane. Are there any existence results for the Dirichlet boundary value problem $$\begin{cases}\Delta f(q) = 0, & q\in \Omega\\ \lim_{p\to q} f(p) = g(q), ...
0
votes
1answer
42 views

Local regularity for harmonic functions (Laplace's equation)

I need a local Sobolev regularity result for a smooth solution $u$ of $$ -\Delta u=0 $$ with the equation satisfied in an open set $U$ (I have no boundary conditions). I know that such a smooth $u$ ...
0
votes
0answers
45 views

Second partial derivatives of harmonic functions

Given a twice-differentiable function $f:S\rightarrow \mathbb{R}$, where $S$ is a nonempty subset of $\mathbb{R}^n$, how does one prove that if f is harmonic then the second partials $D_jD_kf$ all ...
1
vote
1answer
73 views

An application of the strong Markov property in the proof of the connection between Brownian motion and harmonic functions

Let $U$ be an open, connected set in $\mathbb{R}^n$ and let $(B(t))_{t \geq 0}$ be an $n$-dimensional Brownian motion with start at $x \in U$ and let $\overline{B_x(\delta)}$ be the closed ball about ...
3
votes
1answer
115 views

The derivative of harmonic function at origin is a bounded linear functional

The following problem is the 5th problem in the qualifying exam of UCLA (spring 2013). Let $\mathbb{D}=\{(x,y):x^2+y^2<1\}$ and let us define a Hilbert space $$H:=\{u:\mathbb{D} \rightarrow ...
1
vote
2answers
64 views

Harmonic function vanishing on a set of positive measure.

I'm preparing for a qualifying exam, and came across a question I couldn't figure out: If $\Omega$ is a region and $h:\Omega\to \mathbb{R}$ is a harmonic function vanishing on a set of positive ...
1
vote
0answers
23 views

Derivatives of the Green's Function

Consider the function $ g(x,y): \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ defined as $$ g(x,y) = \log( x^2 + y^2) $$ We know that $ (\partial_x^2 + \partial_y^2) g(x,y) = -2 \pi \delta(x)\delta(y) $. ...
3
votes
1answer
60 views

Show $\Omega$ is simply connected if every harmonic function has a conjugate

Prove: If every harmonic function on $\Omega$ has a harmonic conjugate on $\Omega$, then $\Omega$ is simply connected. Same question is asked here but no proof is presented: is the converse true: in ...
1
vote
1answer
90 views

harmonic conjugate of a natural log

Find the harmonic conjugate of $\ln \sqrt{x^2+y^2}$ on some open nonempty subset of the plane. Ok I got stuck. So I set the function as $u(x,y)=\ln \sqrt{x^2+y^2}$. So to show that $u_x(x,y)= ...
0
votes
1answer
37 views

Why do level curves of a function and its harmonic conjugate intersect each other orthogonally?

So I've had this assignment in which I had to proof that two level curves of a function and one of its harmonic conjugates intersect each other orthogonally. The proof itself wasn't that difficult, ...
1
vote
0answers
44 views

Does a complex polynomial can be approximated by a Euler-type exponential function?

Can one uniformly approximate a complex polynomial with a Euler-type exponential function? I mean: let $E\subset\Bbb C$ be a compact set with the connected complement $\Omega:= \bar {\Bbb C}\setminus ...
1
vote
1answer
54 views

Harmonic functions on $\mathbb C-\{0\}$

Find all real-valued $C^2$ differentiable functions $h$ defined on $(0,\infty)$ such that $u(x,y)=h(x^2+y^2)$ is harmonic on $\mathbb C-\{0\}$. This is one of my homework problem. As I understand I ...
0
votes
1answer
33 views

Can we consider the points $(c,y)$ as local minima or maxima for all $y∈ℝ$

Let $f:(a,b)×ℝ→ℝ$ be a non zero and twice differentiable function. Let $c∈(a,b)$. Assume that $Δf=0$ (the Laplacian operator) and $f(c,y)=0$ for all $y∈ℝ$. Assume that $$f(x,y)<0=f(c,y)$$ for ...
0
votes
1answer
39 views

Conformal Mapping and Relating Solutions (of Laplace) of Domains (via the Mapping)

Find a conformal equivalence between the following domains: the strip $ S = \{ z \in \Bbb C \ | \ 0 < \Bbb Im(z) < 1 \} $ and the quadrant $ Q = \{z \in \Bbb C \ | \ \Bbb Re(z) > 0, \Bbb ...
3
votes
0answers
41 views

The Schwarz reflection principle and harmonic function (Big Rudin chapter 11)

In his book page 250 Exer 11: Suppose that $I=[a,b]$,$\Omega$ is a region ,$I\subset\Omega$,$f$ is continuous in $\Omega$,and $f\in H(\Omega-I)$,prove that actually $f\in H(\Omega)$. If I follow the ...
0
votes
1answer
45 views

Limit of natural log

Prove that $\displaystyle \lim_{n \to \infty} \ln x = \infty$ using the fact that the harmonic series diverges Of course, this is obvious graphically, but I have to prove it formally. I based my ...
1
vote
0answers
41 views

Harmonic Function With Step Function Boundary Data

Consider the Unit Disk. can we solve for a harmonic function in the unit disk such that: $\triangle u = 0 $ in D and $ u = f $ on $\partial D$ where $ f = 1$ for $|\theta| \leq \epsilon$ and $ |\theta ...
1
vote
1answer
64 views

Uniqueness for second order PDE $-\Delta u = c(u)$

I'm trying to solve the following problem: Let $\Omega$ be an open set in $\mathbb R^n$ and consider the equation \begin{cases} -\Delta u = c(u) & \text{ on } \Omega \\ u = 0 & \text{ on } ...
1
vote
1answer
30 views

Harmonic inside with zero average

Assume $\Omega\in\mathbb{C}$ is a domain with nice enough boundary,say smooth boundary. What can be said about $f\in C(\bar\Omega)$, harmonic in $\Omega$ and $\int_{\partial\Omega}f(z)|dz|=0$, where ...
2
votes
1answer
74 views

Boundary integral of a harmonic function around a pole

I have a radial harmonic function $h:\mathbb R^N\backslash\{0\}\to\mathbb R$ which has a pole of order $m$ in 0, and I would like to compute $$ \frac{1}{\sigma_N}\int_{\partial ...
1
vote
0answers
225 views

Proof of strong maximum principle for harmonic functions

Let $u\in\mathscr C^2(U)\cap\mathscr C(\bar U)$ be harmonic in the non-empty open and connected set $U\subset\mathbb R^n$. If there exists a Point $x_0\in U$, so that $u$ has a local Maximum at $x_0$, ...
2
votes
2answers
33 views

Let $F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha$. Is it true $F, F^{-1}\in L^{1}(\mathbb R)$?

Define $$F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha, \ (x\in \mathbb R).$$ It is clear to me that, the integral converges for every real $x$ (as near origin integrand is ...
2
votes
1answer
148 views

Show that $f$ is harmonic

Let us consider the function: $$ f(α,β) \equiv \sum_{n = 1}^{\infty}\left(-1\right)^{n - 1}\left[% {n^{2\alpha - 1} - 1 \over n^{\alpha}}\,\cos\left(\beta\ln\left(n\right)\right) \right] $$ My ...
3
votes
2answers
113 views

If a Laplacian eigenfunction is zero in an open set, is it identically zero?

This seems like it should be easy but I can't seem to figure it out. Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set ...
1
vote
0answers
45 views

How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)?

We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and ...
0
votes
1answer
63 views

Function must be constant comparison

The following 2 problems are past exit exam problems for my major. I see that they're worded differently but are asking me to do the same thing. Not sure how they differ much I'd appreciate if anyone ...
1
vote
0answers
47 views

Biharmonic boundary condition

I try to solve $$\Delta^2u=f$$ on unit square. with $f=4sin(\pi x)sin(\pi y)$ Using $v=-\Delta u,$ leads to $$v+\Delta u=0,$$ $$-\Delta v=f.$$ By Dirichlet boundary condition on $u$. What is ...
0
votes
0answers
31 views

Harmonic function [duplicate]

Let $B(0;1)=\{x \in \mathbb{R}^N;|x|≤1\}$, the ball of $\mathbb{R}^N$ equipped with the euclidian scalar product $$x⋅y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ \ y=(y_1,...,y_N)\ \ \ |x|=\sqrt{x ...
0
votes
1answer
111 views

Choose parameters to make a harmonic function

Let $B(0;1)=\{x\in \mathbb{R}^N ;|x|\leq 1\}$, the ball of $\mathbb{R}^N$ equipped with the euclidian scalar product $$x \cdot y=x_1y_1+...+x_Ny_N,\ \ \ x=(x_1,...,x_N),\ \ y=(y_1,...,y_N)\ \ \ ...
0
votes
1answer
34 views

$\{u_{n}\}$ harmonic and converging uniformly to $u \Rightarrow $ $u$ harmonic

Let $A$ be an open set, $\{u_{n}\}$ a sequence of harmonic functions on $A$, converging to $u$ uniformly on compact subsets of $A$ . Then $u$ is harmonic on $A$. Any hint ?
5
votes
1answer
191 views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
0
votes
2answers
58 views

implicit derivates incorperating laplace's equation

If $f(x,y)$ is a harmonic function show that the function $F(x,y)=f(x^2-y^2,2xy)$ is also harmonic. You have to use Laplace's formula to prove this, unless there is an easier way. I'm having trouble ...
1
vote
2answers
49 views

Harmonic functions constant on circumferences

I want to find all the harmonic functions in $\mathbb{R}^{2}-\{(0,0)\}$ which are constant on circumferences with center in $(0,0)$. $\mathbb{R}^{2}-\{(0,0)\}$ isn't simply connected so we can't ...
0
votes
1answer
48 views

Bounded (from below) harmonic functions from $\mathbb R^2 \setminus \{0\}$

Let $M \in \mathbb R$ be a real number and $u\colon \mathbb R^2 \setminus \{0\} \to [M, +\infty)$ be an harmonic function. Then it is constant. Show that this is no longer true in higher ...
1
vote
1answer
32 views

Examples for 2-dimensional real valued harmonic functions

Given: $$f: \mathbb{R}^2 \to \mathbb{R},\space \Delta{f}=0, \space\frac{\partial^2{f}}{\partial{x}^2_i} \neq 0, i \in\{1,2 \}.$$ Are there examples of such functions?
1
vote
1answer
44 views

One problem about harmonic functions

Problem. Given open, bounded set $\Omega\subset\mathbb R^d$ with smooth boundary $\partial\Omega$ and given smooth function $\varphi$ on $\partial\Omega$. As known, problem $$ \begin{cases} ...
3
votes
2answers
102 views

$uv$ is harmonic if and only if $u+icv$ is analytic for some real c

Let $u$ and $v$ be non constant harmonic functions on a complex domain. Prove that $uv$ is harmonic if and only if $u+icv$ is analytic for some real $c$. I can prove the "if" part. I am having some ...
8
votes
1answer
250 views

Error on Wikipedia: Nelson's proof of Liouville's theorem works only for bounded modulus?

On Wikipedia, it is stated: If $f$ is a harmonic function defined on all of $\mathbb{R}^n$ which is bounded above or bounded below, then $f$ is constant...Edward Nelson gave a particularly ...
1
vote
0answers
45 views

What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...
5
votes
4answers
159 views

Compute $\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right )^2+\left (\frac{1}{n} \right )^2$

Compute the value of the following expression $$\left (1+\frac{1}{2}+\cdots+\frac{1}{n} \right )^2+\left ( \frac{1}{2}+\cdots + \frac{1}{n}\right )^2+\cdots+\left (\frac{1}{n-1}+\frac{1}{n} \right ...